Effect of signal discreteness on correlation functions

Number or density autocorrelation functions can sometimes be obtained in terms of electron emission current autocorrelation functions if the current fluctuations are related to the number fluctuations. This is the case in field emission where the fluctuations in adsorbate number in a small probed region, caused by surface diffusion, are essentially proportional to the field emission current fluctuations. The autocorrelation function of the current fluctuations then mirrors that of the adsorbate number fluctuations. When correlation intervals are very short they may contain an average number of electrons so small that the rms electron fluctuation resulting from adsorbate number fluctuations is less than 1 e− and thus shows up only infrequently. The electron fluctuation can then ‘‘miss’’ most adsorbate fluctuations. We have examined this problem by Monte Carlo modeling and confirm the existence of this effect when the electron fluctuation number is truncated to integer values. Even in this case the electron fluctuation correlation images the adsorbate number correlation function quite well to values of rms electron fluctuations as low as 0.4 e− per correlation interval. When shot noise is included imaging is good to the lowest value tested, 0.18 e−, but the number of summations required to overcome noise goes up very steeply. The reason for the improvement in imaging is that shot noise circumvents the problem of electron fluctuation inadequacy, since very small changes in mean electron emission probability, corresponding to much less than 1 e− per correlation interval, are reflected in the Poisson distribution. Thus shot noise in a way ‘‘amplifies’’ the fluctuations but at the price of vastly increased noise. This conclusion is probably not confined to the specific case examined here but seems generally valid.